Substitutions over Infinite Alphabet Generating (-β)-Integers

This contribution is devoted to the study of positional numeration systems with negative base introduced by Ito and Sadahiro in 2009, called (−β )-expansions. We give an admissibility criterion for more general case of (−β )-expansions and discuss the properties of the set of (−β )-integers, denoted by Z−β . We give a description of distances within Z−β and show that this set can be coded by an infinite word over an infinite alphabet, which is a fixed point of a non-erasing non-trivial morphism.